Darko Babic

The Laplacian matrix in chemistry

The Laplacian matrix, its spectrum, and its polynomial are discussed. An algorithm for computing the number of spanning trees of a polycyclic graph, based on the corresponding Laplacian spectrum, is outlined. Also, a technique using the Le Verrier-Faddeev-Frame method for computing the Laplacian polynomial of a graph is detailed. In addition, it is shown that the Wiener index of an alkane tree can be given in terms of its Laplacian spectrum. Two Mohar indices, one based on the Laplacian spectrum of a molecular graph G and the other based on the Laplacian x2 eigenvalue of G, have been tested in the structure-property relationships for octanes.

Partial Orderings in Chemistry

The cosmopolitan relevance of partially ordered mathematical structures in chemistry is argued. Many examples are briefly noted, including those involving chemical periodicities, reactivities, aromaticities, electronegativities, molecular branching, molecular shapes, symmetries, complexities, curve fittings, and more. A few fundamental theorems concerning metrics (or distance functions) on partially ordered sets are noted, first for the intuitively appealing “scaled” posets, then for the more general “transformed” posets. Interspersed along the way are a few examples which are developed to a greater extent, including Randic−Wilkins periodicity for alkanes; the general concept of aromaticity; molecular branching; least-squares fittings; and (most extensively) molecular shapes, chiralities, and symmetries. In each of these types of examples clarifications, alternative views, and extensions of previous works result.

Pyracylene rearrangement classes of fullerene isomers

Abstract Isomers of C n , n ⩽ 70, were generated and classified according to their interconvertibility by the pyracylene rearrangement. The algorithm used for generation of fullerene isomers represents an improvement of the spiral ring algorithm. Class cardinalities and the resonance energies of the most stable representatives are tabulated.

Topological Resonance Energy of Fullerenes

A new algorithm for computation of a matching polynomial is outlined. It has been successfully applied for evaluation of the topological resonance energy (TRE) of all fullerene isomers with up to, and including, 70 carbon atoms. The obtained results indicate a high correlation between TRE values and π-electronic energies (Eπ), which was found to be a consequence of the invariance of the reference energy values for a given size of isomers. A similar high correlation was established between TRE and Eπ values normalized to a number of carbon atoms, thus extending over different sizes of fullerene isomers. These results show that the resonance energy does not bring any new insight into stabilities of different fullerene isomers which is not already achieved by consideration of Eπ.

Mo¨bius inversion on a poset of a graph and its acyclic subgraphs

Abstract Several chemical models for resonance energy of conjugated hydrocarbons are formulated in terms of a graph and its acyclic subgraphs. The resonance energy in these models is expressed by a truncated cluster expansion in terms of acyclic subgraphs. In order to efficiently perform the complete expansion, one needs to know the Mobius function for the pertinent partially ordered set. In the present paper a formula for the Mobius function in the poset is derived and discussed.

Benzenoid graphs with equal maximum eigenvalues

The authors stated that according to their calculation the maximum eigenvalues of G1 and G2 could be equal. Since we are in a position to carry out the diagonalization of the adjacency matrix to high accuracy, we found the answer: Benzenoid graphs G1 and G2 do not possess equal maximum eigenvalues. The calculated maximum eigenvalues are as follows: Xma x ( G 1 ) = 2.6369724379565793242087106873803083111541 and Xmax (G2) = 2.6369724369094896734421158581663075141815.